So, did you see that last week some lucky person in South Carolina won the $400 million Powerball Jackpot? (Technically, it was $399.8 million, but $400 million is close enough!)
It seems like every time there’s a big jackpot won in the lottery you read statements like “you’re more likely to be struck by lightening,” or “you’re more likely to marry a prince,” or in the case of the CNN story about this particular Powerball winner “you’re more likely to get struck by lightening and bitten by a shark.” (Talk about a bad day!) They also go an to say that the chances of winning a Powerball Jackpot are 1 in 175223510. Don’t you wonder how people come up with all of these statistics?
Let’s take a look at how Powerball is played . . .
According to the Powerball website, lottery numbers are drawn from two drums. The first drum contains 59 white balls and the second drum contains 35 red balls (These red balls are all potential Powerballs). The jackpot is won by matching all five white balls in any order and the red Powerball.
In order to calculate the odds of winning, we need to figure out the odds of matching all five white balls, in any order, and the red ball.
Let’s start with the white balls:
I like to think about situations like the one described above by picturing an empty (in this case) lottery ticket. Like this:
The first ball that pops up could be any of the 59 balls, the second ball could be any of the 58 balls, the remaining blank spots on the ticket will be filled by drawing from the final 57, 56, and 55 balls respectively.
That looks like this:
Now, the order doesn’t matter in the way I arrange the balls, remember? That means if the winning white balls are 1, 2, 3, 4, 5 and my ticket is 2, 3, 4, 1, 5; I’m on my way to winning the Powerball! So now we need to figure out how many different ways the 5 white ball numbers can be arranged.
The first white number could be any of the 5 numbers drawn from the drum, so I have 5 choices for the number in the first spot. I only have 4 remaining numbers for the second spot, 3 for the 3rd spot, etc . . .
Because I can rearrange the numbers and still have a winning ticket, the possible number combinations I need to win has just been decreased! Now, we can calculate the number of ways to get a winning combination from the white balls in the Powerball drawing:
The total number of combinations is :
And now for the red Powerball! The process we used above is going to be the same for red Powerball, except instead of having to match 5 numbers you only have to match 1 and because there’s only one number to be matched it doesn’t really make sense to talk about whether or not the order matters . . . there’s only one number.
Since you know a method to use and you know the answer, I’m pretty confident you are going to be able to figure how CNN could report that a person has a 1 in 175223510 chance in winning.
P.S. I used a few different techniques from Discrete Mathematics or Counting Theory in this post that I didn’t explicitly name. First, as in the case of the white lotto balls I was choosing 5 balls from a collection of 59 balls. The order in which I arranged these 5 balls was not important. This situation describes a combination or a binomial coefficient. There is a formula associated with these types of situations. We used this formula, although I did lots of canceling to make the numbers used look less overwhelming. You could rewrite the situation that I described above as a combination using the formula provided from the combination link and see if you can get it to look like the one I used.
Also, we calculated the factorial of 5, denoted 5!. If you attempt the challenge I’ve given you above, you’ll want to make sure you know what a factorial is.